When Conan first fights Remo, Conan places a blood stain on his forehead by pressing his sword to it. Remo then runs off, and Conan gives chase immediately, pausing only to get Tamara on his horse. The stain is still there when Conan mounts his horse. The next time we see them, Conan and Tamara is still chasing Remo, but now Conan's face is completely clean. It is highly unlikely he would pause in pursuit of one of his father's killers to wash his face.